Step by step
The question
Question: A farmer needs 40 litres of a 35% feed mixture.
He mixes a 20% feed mixture with a 60% feed mixture.
How many litres of each does he use?
1 Find what we know
What do we know?
The mix: 40 litres that is 35% feed.
It is made from a 20% and a 60% mixture.
What do we want?
How many litres of each.
2 Name the unknown
We don't know how many litres of the 60% mixture.
So we call it x.
The rest is the 20% mixture.
40 litres in all, take away x: 40 − x.
Let's put it in a table: how much, the feed part as a decimal, and the feed.
60% is 0.6.
20% is 0.2.
Feed = how much times the part: 0.6x.
0.2 times all of 40 − x: 0.2(40 − x).
We keep 40 − x together in brackets.
The mix: 40 litres at 0.35.
40 times 0.35 is 14 litres of feed.
3 Write the equation
In words: the feed from the two mixtures makes the feed in the mix.
So the equation is 0.6x + 0.2(40 − x) = 14.
Now we solve it.
Careful!
Many people take half of each.
But half and half gives (60 + 20) ÷ 2 = 40%, not 35%.
4 Remove the decimals
Decimals are hard to work with.
Let's get rid of them.
We multiply every part by 10.
We do it on both sides, so they stay equal.
Times 10 moves the point 1 place to the right.
10 × 0.6x = 6x.
10 × 0.2(40 − x) = 2(40 − x).
10 × 14 = 140.
Write the answers.
No more decimals!
5 Open the brackets
The 2 in front of the bracket multiplies every part inside.
So we write 2 times each part.
Work out each multiplication.
6 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
6x − 2x = 4x.
Now we have 4x + 80 = 140.
We want x by itself.
Two things are stuck to x: times 4 and + 80.
The + 80 is on the outside, so it comes off first.
Then the times 4.
7 Remove the + 80
To remove + 80, we take away 80.
We do it on both sides, so they stay equal.
+ 80 and − 80 make zero.
Cross them out.
140 − 80 = 60.
Now we have 4x = 60.
8 Remove the 4
4x means 4 times x.
We want just x.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just x is left.
60 divided by 4 is 15.
So x = 15!
9 Answer the question
The 20% mixture: 40 − 15 = 25.
So 15 litres of the 60% mixture and 25 litres of the 20% mixture.
10 Check
Let's check in the story.
15 litres and 25 litres make 40 litres, like the story says.
The feed: 9 plus 5 is 14 litres.
That is 35% of 40, like the story says.
Does it make sense?
The mix is between the two mixtures.
So the answer is right!
Let's look back.
First: find what we know.
Then: name the unknown.
Then: write the equation.
Then: solve it.
Then: answer the question.
Last: check.